Page 41 - Geometric Modeling and Algebraic Geometry

38 F. Aries et al.
Symbol Name Degree Type
Invariants C
∆ Discriminant 24 C
Families of objects
Tangent plane at the image of a point 3 Pol (C , (C ) )
4 ∗
3
Φ 1
3
Linear plane spanned by the image of a line 3 Pol ((C ) , (C ) )
3 ∗
4 ∗
Φ 2
3
3 ∗
Correspondence line–line 4 Pol ((C ) , (C ) )
3 ∗
Φ 3
2
Preimage of a point on S(f) 10 Pol (C , C )
4
3
Φ 6
2
Associated surfaces in CP 3 Pol (C )
4
n
Implicit Equation 12 n =4
Φ 4
Associated Quadric 6 n =2
Φ 5
Union of the Tropes 12 n =4
Φ 9
Trihedron defined by the Double Lines 21 n =3
Φ 10
Polar Plane Π of the Associated Quadric and the 15 n =1
Φ 12
Triple Point
Associated surfaces in (CP ) Pol ((C ) )
4 ∗
3 ∗
n
Dual surface 3 n =3
Φ 7
Triple Point 9 n =1
Φ 8
Associated curves in CP 2 Pol (C )
3
n
Exceptional Triangle 12 n =3
Φ 11
Conic preimage of Π 16 n =2
Φ 13
Quadrilateral preimage of the torsal conics 8 n =4
Φ 15
3 ∗
2 ∗
Associated surfaces of (CP ) Pol ((C ) )
n
Dual conic to the preimage of Π 8 n =2
Φ 14
Table 2.2. List of the covariants presented in the paper.

∂ 0 f 0 ∂ 1 f 0 ∂ 2 f 0 y 0
1
Φ 1 = ∂ 0 f 1 ∂ 1 f 1 ∂ 2 f 1 y 1 . (2.12)

8 ∂ 0 f 2 ∂ 1 f 2 ∂ 2 f 2 y 2
∂ 0 f 3 ∂ 1 f 3 ∂ 2 f 3 y 3
Here ∂ i stands for ∂ .
dx i
This covariant Φ 1 has degree 3 and type Pol (C , (C ) ). The geometric object
4 ∗
3
3
associated to Φ 1 (f) is a parameterization of the dual surface to S(f).
Plane spanned by the image of a line.
Consider a generic line L in CP , given by an equation
2
λ(x)= λ 0 x 0 + λ 1 x 1 + λ 2 x 2 =0. (2.13)
3 ∗
Its image under f is a conic in CP , spanning a plane, that is an element of (CP ) .
3
This plane is always a tangent plane to S(f). It admits Φ 2 (f)(λ)=0 as an equation,
with